Theoretical Economics Letters, 2012, 2, 351-354

http://dx.doi.org/10.4236/tel.2012.24064 Published Online October 2012 (http://www.SciRP.org/journal/tel)

On the Closed-Form Solution to the Endogenous Growth

Model with Habit Formation

Ryoji Hiraguchi

Faculty of Economics, Ritsumeikan University, Kyoto, Japan

Email: rhira@fc.ritsumei.ac.jp

Received June 25, 2012; revised July 23, 2012; accepted August 20, 2012

ABSTRACT

We study the AK growth model with external habit formation. We show that there exists a unique solution path expressed

in terms of the Gauss hypergeometric function. Using the closed-form solution, we also show that the opti mal path con-

verges to a balanced growth path.

Keywords: Endogenous Growth; Closed-Form Solution; Habit Formation

1. Introduction

The Gauss hypergeometric functions are typically used in

mathematical physics, but are not so common in eco-

nomics. As far as we know, Boucekkine and Ruiz-

Tamarit [1] are the first to find that the functions are also

useful in dynamic macroeconomics. They obtain an ex-

plicit solution path to Lucas-Uzawa two-sector endoge-

nous growth model by using the hypergeometric function.

They express the optimal path as a system of four dif-

ferential equations and two transversality conditions and

then use the hypergeometric functions for solving the

system of equations.

Several authors get analytical solution paths to the

exogenous growth models. Pérez-Barahona [2] investi-

gates the model with non-renewable energy resources

and find that the optimal path has a closed form solution

path by using the hypergeometric function. Hiraguchi [3]

finds that a solution path to the neoclassical growth

model with endogenous labor is also represented by the

special function.

Boucekkine and Ruiz-Tamarit [1] argue (see page 34)

that the hypergeometric functions will also be useful in

the investigation of the endogenous growth models.

They guess that the transition dynamics of the models

will be easier to understand if we can use the special

functions. However, there is only a few literature that

applies the special functions to the endogenous growth

models other than the Lucas-Uzawa model. One ex-

ample is Guerrini [4] who uses the special functions and

obtains a closedform solution path to the AK model

with logistic population growth. Broad applicability of

the hypergeometric functions to the endogenous growth

models is uncertain at this point, and more investiga-

tions are needed.

In this paper, we study the AK endogenous model with

external habit formation. The model has been investi-

gated by many authors including Carroll et al. [5] and

Gómez [6]. The utility function of the agent depends on

both the absolute level of consumption and the ratio be-

tween consumption and habit stock. Here the habit for-

mation is external and the level of the habit stock is ex-

ogenous to each agent. We show that there exists a unique

solution path and it is represen ted by the hypergeometric

function.

Habit formation in consumption is now popular in

modern macroeconomics. Authors have explained some

empirical facts by incorporating habits into the dynamic

macroeconomic models. Abel [7] and Gal [8] show that

habit formation can solve the equity premium puzzle in

asset pricing models and Carroll et al. [9] provide an

explanation of strong correlations between saving and

growth. Some authors characterize the properties of the

optimal paths in these models. Alvarez-Cuadrado et al.

[10], Alonso-Carrera et al. [11] and Gómez [6] in-

vestigate the transitio nal dyn amics and the stability of the

optimal paths in endogenous growth models with habits,

both analytically and numerically.

The problem of the previous papers is that they assume

the existence and the uniqueness of the optimal path

without proof. These properties are not at all obvious

here, because there exists no general theorem on the

existence of a solution path in an infinite horizons opti-

mization problem with externalities. Here we utilize the

special functions to show that their assumptions are in

fact correct.

The note is organized as follows. Section 2 describes

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